The local density approximation (LDA) requires that the exchange-correlation potential be given as a function of the electron density at a given point in space. For this part of our study, we use the form of the exchange-correlation potential given by Vosko, Wilk, and Nusair (VWN). The form is a fit to the Ceperley-Alder electron gas study . The VWN functional reproduces the random phase approximation (RPA) results for a uniform electron gas in the high density limit, it reproduces the spin-stiffness constant calculated in the RPA in the paramagnetic limit of a uniform electron gas, and it is uniformly differentiable as a function of the electron gas parameter, . It is also in standard use, or available as an option, in many electronic structure codes, and thereby provides a convenient reference potential for checking the accuracy of numerical calculations.
We now summarize the form of the VWN functional.
The exchange-correlation energy per electron is separated into two parts, an exchange term and a correlation term.